2.5
Proving
Statements about
Segments
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Objectives
• Students will justify statements about
congruent segments.
• Students will write reasons for steps in a
proof.
Properties of Congruent
Segments
• Theorem – a true statement that follows as
a result of other true statements
• Two – column proof – numbered
statements and reasons that show the
logical order of an argument
• Paragraph Proof – a proof written in
paragraph form (see page 102)
Theorem 2.1 Properties of
Segment Congruence
•
•
•
•
Reflexive: For any segment AB, AB  AB
Symmetric: If AB  CD , then CD  AB
Transitive: If AB  CD and CD  EF, then AB  EF
two-column proof – numbered statements
and reasons to show their logical order
Example 1
Given: EF=GH Prove: EG  FH
E
F
G
H
Statements
Reasons
1. EF=GH
2. EF+FG=GH+FG
3. EG=EF+FG,
FH=FG+GH
4. EG=FH
5. EG  FH
1. Given
2. Add. Prop. of Eq.
3. Segment Add. Post
4. Substitution
5. Def. of  segments
Example 2
R
S
T
Given: RT  WY , ST  WX Prove: RS  XY
W
Statements
Reasons
1.
2.
3.
4.
5.
6.
7.
1.
2.
3.
4.
5.
6.
7.
RT  WY
RT=WY
RT=RS+ST, WY=WX+XY
RS+ST=WX+XY
ST=WX
RS=XY
RS  XY
X
Y
Given
Def. of  segments
Segment Add. Post.
Substitution
Given
Subtraction Prop. of Eq.
Def. of  segments
Example 3
S
M
Given: X is the midpoint of MN and MX=RX Prove: XN=RX
X
R
Statements
1.
2.
3.
4.
X is the midpoint of
XN=MX
MX=RX
XN=RX
Reasons
MN
1.
2.
3.
4.
Given
Def. of  segments
Given
Substitution
N
More Example….
See book pages 102-103 for more PROOFS
Symmetric Property of Segment
Congruence
Using Congruence
Using Segment Relationships
Practice using a compass to copy a
segment